Deformations of semi-direct products

Abstract

We exhibit in this article a contraction of the direct product Lie algebra $g\oplus g$ of a finite-dimensional complex Lie algebra $g$ onto the semi-direct product Lie algebra $g⋊g$, where the first factor $g$ is viewed as a trivial Lie algebra and as the adjoint $g$-module. This contraction gives rise to a non-zero cohomology class in the second cohomology space. We generalize to the setting of $h\oplus g$ and $h⋊g$ with respect to a given crossed module of Lie algebras $h\to g$. We give many examples to illustrate our results.

19 pages, no figures